lab_quacks Spiteful Stacks and Questionable Queues

Assignment Description

In this lab, you will learn to think recursively and apply it to the
stack and queue data structures. You might also practice templates.

Lab Insight

Stacks and queues are incredible data structures used in a wide range of applications throughout the world. You have already seen a great example of a queue in CS 225. We use the queue data structure to help us manage our office hours. Queues can do so many incredible things like helping schedule tasks for a computer or managing user priority based on a FIFO (first in, first out) principle. You will find stacks useful in classes like CS 438, Communication Networks. You will find queues useful in CS 241, Systems Programming. In general, these data structures are very versatile and useful throughtout the software world.

Recursion

What is recursion? Recursion is a way of thinking about problems that
allows the computer to do more of the heavy lifting for us. It is analogous to
the mathematical definition of recursive functions, where you can define a function call
in terms of calls to itself and basic arithmetic operations, but not in
terms of loops.

Why recursion? While being able to think recursively is one of the harder
parts of computer science, it is also one of the most powerful. In fact, there
are whole languages that entirely use recursion instead of loops, which, even
though it may seem inefficient, leads to some very useful optimizations a
compiler can make when dealing with such code. There are probably more problems
in computer science that are simpler recursively than they are iteratively
(using loops). Also, once you have a recursive algorithm, it is always possible
to transform it into an iterative algorithm using a stack and a while loop. In
this way, computer scientists can think about problems recursively, then use
that recursive solution to make a fast iterative algorithm (and in the grand
scheme of big-O notation, using recursion has little overhead compared to the
rest of the running time). Here we’ll only ask you to do the first part.

How do I write recursively? Recursion just means calling a function within
itself. This may sound crazy, but in fact it is not. Let’s take
an iterative function to calculate the factorial of a number \(n\), \(n!\):

Only two lines of code! (Depending on whether you like putting your return
statement on the same line.) Even on such a small problem, recursion helps us
express ourselves more concisely. This definition also fits better with the
mathematical definition:
\[
n! = \begin{cases}
1 & \text{if $n = 0$,} \\
(n-1)!\times n & \text{if $n > 0$.}
\end{cases}
\]

A typical recursive function call consists of three parts. Let’s examine the
function more closely to see them. Here’s the same code again, with more
discussion.

Example

int factorial(int n)
{
if (n == 0) // Here is our base case.
return 1; // The base case is the smallest problem we can think of,
// one we know the answer to. This is the "n = 0" case in
// the mathematical definition.
// optional 'else' here
return (factorial(n-1) // This is our recursive step. Here we are solving a
// smaller version of the same problem. We have to
// make a leap of faith here - trust that our
// solution to the (n-1) case is correct. This is
// the same as the mathematical definition, where
// to figure out n!, we need to first figure out
// (n-1)!
* n // Here is our incremental step. We are transforming
// our solution to the smaller problem into the
// solution to our larger problem. This is the
// same * n from the mathematical definition.
);
}

Checking Out the Code

After reading this lab specification, the first task is to check out the
provided code from the class repository.

To check out your files for the third lab, run the following command in your
cs225git directory:

This should update your directory to contain a new directory called
lab_quacks.

STL Stack and Queue

These activities use the standard template library’s stack and queue
structures. The interfaces of these abstract data types are slightly different
than in lecture, so it will be helpful for you to look up “STL Stack” and “STL
Queue” on Google (C++ reference has good information). In particular, note that
the pop() operations do not return the element removed, and that you must
look that up before calling pop().

Recursive Exercises

No loops!

You may not use any loops for this section! Try to think about the problem
recursively: in terms of a base case, a smaller problem, and an incremental
step to transform the smaller problem to the current problem.

Compute recursively (no loops or multiplication) the total number of blocks in
such a triangle with the given number of rows.

int triangle(int rows);

triangle(0) -> 0
triangle(1) -> 1
triangle(2) -> 3

Note

These examples were stolen from http://codingbat.com/java/Recursion-1. All
credit goes to CodingBat. If you are having a hard time with sum (below), we
encourage you to go to CodingBat and try more recursive exercises. These are in
Java, but there are links at the bottom of the page describing the differences
of strings and arrays in Java from C++, which are minor.

The sum Function

Write a function called sum that takes one stack by reference, and returns
the sum of all the elements in the stack, leaving the original stack in the
same state (unchanged). You may modify the stack, as long as you restore it to
its original values. You may use only two local variables of type T in your
function. Note that this function is templatized on the stack’s type, so stacks
of objects overloading the addition operator (operator+) can be summed. Hint:
think recursively!

STL Stack

We are using the Standard Template Library (STL) stack in this problem. Its
pop function works a bit differently from the stack we built. Try searching
for “STL stack” to learn how to use it.

template <typename T>
T QuackFun::sum(stack<T> & s);

Non Recursive Exercises

Balancing Brackets: the isBalanced Function

For this exercise, you must write a function called isBalanced that takes one
argument, an std::queue, and returns whether the string represented by the
queue has balanced brackets. The queue may contain any characters, although you
should only consider square bracket characters, ‘[’ and ‘]’, when considering
whether a string is balanced. To be balanced, a string must not have any unmatched,
extra, or hanging brackets. For example, the string [hello][] is balanced,
[[][[]a]] is balanced, []] is unbalanced, ][ is unbalanced, and ))))[cs225]
is balanced. It’s possible to solve this problem without using a stack, but in the
spirit of this lab, you should use one in your solution!

For this function, you may only create a single local variable of type stack<char>! No other stack or queue local objects may be declared.

bool isBalanced(queue<char> input);

The scramble Function

Your task is to write a function called scramble that takes one argument: a
reference to a std::queue.

template <typename T>
void QuackFun::scramble(queue<T> & q);

You may use whatever local variables you need. The function should reverse the
order of SOME of the elements in the queue, and maintain the order of others,
according to the following pattern:

The first element stays on the front of the queue.

Then the next two elements are reversed.

Then the next three elements are placed on the queue in their original order.

Then the next four elements are reversed.

Then the next five elements are place on the queue in their original order.

etc.

Hint: You’ll want to make a local stack variable.

For example, given the following queue,

front back
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

we get the following result:

front back
0 2 1 3 4 5 9 8 7 6 10 11 12 13 14 16 15

Any “leftover” numbers should be handled as if their block was complete. (See
the way 15 and 16 were treated in our example above.)

STL Queue

We are using the Standard Template Library (STL) queue in this problem. Its
pop function works a bit differently from the queue we built. Try searching
for “STL queue” to learn how to use it.

Good luck!

(Extra-Credit) The verifySame function

Complier errors

Submitting code that doesn’t compile will result in a zero on the entire
lab. This includes code in the extra credit portion and should be common sense.
Be sure to test your code before you submit.

Extra Credit

This function is NOT part of the standard lab grade, but is extra credit.
It was also a previous exam question, and something similar could show up
again.

Write the recursive function verifySame whose function prototype is below.
The function should return true if the parameter stack and queue contain
only elements of exactly the same values in exactly the same order, and false
otherwise (see example below). You may assume the stack and queue contain the
same number of items!

We’re going to constrain your solution so as to make you think hard about
solving it elegantly:

Your function may not use any loops;

In your function you may only declare ONE local boolean variable to use in
your return statement, and you may only declare TWO local variables of
parametrized type T to use however you wish. No other local variables can be
used; and

After execution of verifySame, the stack and queue must be unchanged. Be
sure to comment your code VERY well.

Example

This stack and queue are considered to be the same. Note that we match the
bottom of the stack with the front of the queue. No other queue matches this
stack.